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Numerical evaluator of integrals implementing the "Sector Decomposition" method (see arXiv:1703.09692, arXiv:hep-ph/0004013, arXiv:0803.4177).

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pySecDec is a toolbox for the calculation of dimensionally regulated parameter integrals using the sector decomposition approach.

See 2108.10807, 1811.11720, and 1703.09692 for description of the implementation; 0803.4177 and hep-ph/0004013 for description of sector decomposition; and 1502.06595 for SecDec, the predecessor of pySecDec.


pySecDec should work under Python version 3.6 or newer on the usual Unix-like systems.

The latest release can be installed from PyPI by first upgrading pip:

$ python3 -m pip install --user 'pip>=20.1'

and then running:

$ python3 -m pip install --user --upgrade pySecDec

The geometric method and Normaliz

If you want to use the geometric decomposition methods (decomposition_method='geometric' or 'geometric_ku'), you need the normaliz executable, which can be downloaded from [W. Bruns and B. Ichim and T. Römer and R. Sieg and C. Söger]. The geometric decomposition module is designed for normaliz version 3 - currently versions 3.3.0, 3.4.0, 3.5.4, 3.6.0, 3.6.2, 3.7.3, 3.7.4, and 3.8.1 are known to work. We recommend to set your $PATH such that the normaliz executable is found. Alternatively, you can pass the path to the normaliz executable directly to the functions that need it.

Additional dependencies for generated C++ packages

The intended main usage of pySecDec is to make it write C++ packages using the functions pySecDec.code_writer.make_package and pySecDec.loop_integral.loop_package. In order to build these C++ packages, the following additional non-python-based libraries and programs are required:

The functions pySecDec.code_writer.make_package and pySecDec.loop_integral.loop_package can use the external program dreadnaut to find all sector symmetries and therefore reduce the number of sectors:

  • NAUTY ( [B. D. McKay and A. Piperno, Practical graph isomorphism, II, 2014, Journal of Symbolic Computation, 60, 94-112, doi:10.1016/j.jsc.2013.09.003]

These packages are redistributed along with pySecDec itself, and will be built automatically during pySecDec installation.

Basic Usage

You can find the pySecDec manual over at Additionally, the development repository contains a folder examples with examples of pySecDec usage.

A simple example of the evaluation of a loop integral with pySecDec is box1L. This example computes a one-loop box with one off-shell leg (with off-shellness s1) and one internal massive line (with mass squared msq).

To run this example change into the box1L directory and run the commands:

$ python3
$ make -C box1L
$ python3

The file defines the loop integral and calls pySecDec to perform the sector decomposition. Once run it produces the directory box1L which contains the code required to numerically evaluate the integral. The make command builds this code and produces a library. The file loads the integral library and evaluates the integral for a specified numerical point.

This will print the result of the integral evaluated with Mandelstam invariants s=4.0, t=-0.75 and s1=1.25, msq=1.0:

eps^-2: -0.142868356275422825 - 1.63596224151119965e-6*I +/- ( 0.00118022544307414272 + 0.000210769456586696187*I )
eps^-1: 0.639405625715768089 + 1.34277036689902802e-6*I +/- ( 0.00650722394065588166 + 0.000971496627153705891*I )
eps^0 : -0.425514350373418893 + 1.86892487760861536*I +/- ( 0.00706834403694714484 + 0.0186497890361357298*I )


Several examples of pySecDec usage can be found in the source code repository (examples directory). Here are some of them:

  • easy: a simple parametric integral, described in the manual in Section 2.1.
  • box1L: a simple 1-loop, 4-point, 4-propagator integral, described in the manual Section 2.2.
  • triangle2L: a 2-loop, 3-point, 6-propagator diagram, also containing massive propagators.
  • box2L_numerator: a massless planar on-shell 2-loop, 4-point, 7-propagator box with a numerator, either defined as an inverse propagator or in terms of contracted Lorentz vectors
  • triangle3L: a 2-loop, 3-point, 7-propagator integral, demonstrates that the symmetry finder can significantly reduce the number of sectors.
  • elliptic2L_euclidean: an integral known to contain elliptic functions, evaluated at a Euclidean phase-space point.
  • elliptic2L_physical: an integral known to contain elliptic functions, evaluated at a physical phase-space point.
  • triangle2L_split: a 2-loop, 3-point, 6-propagator integral without a Euclidean region due to special kinematics.
  • hypergeo5F4: a general dimensionally regulated parameter integral, corresponding to a Hypergeometric function 5F4.
  • 4photon1L_amplitude: calculation of the 4-photon amplitude, showing how to use pySecDec as an integral library in a larger context.
  • two_regulators: an integral involving poles in two different regulators.
  • userdefined_cpp: a collection of examples demonstrating how to combine polynomials to be decomposed with other user-defined functions.


During development instead of full installation it is more convenient to work directly from the checked out repository. To make this work, one needs to install the Python dependencies, build the contributed software, and setup PYTHONPATH to point to the sources; do this by running

$ make dependencies
$ make build

The Makefile in the package's root directory also implements other common development tasks. You can list all available targets with the command

$ make help

pySecDec comes with a self test suite written in the python unittest framework. The most convenient way to run all test is using Nose. If Nose is installed (as it would be after make dependencies), type

$ make check

in the source repository to run all tests. Developers should write test cases for ALL functions they implement and make sure that ALL tests pass before uploading a commit.

Building the Documentation

To build the documentation of pySecDec, you need Sphinx. If Sphinx is installed (as it would be after make dependencies), the command

$ make doc

generates the documentaion in html and in pdf format. Developers should inline-document python functions and also keep the C++ part up to date. To generate HTML and PDF separately, use

$ make doc-html


$ make doc-pdf

Building the documentaion in pdf format requires an up-to-date installation of a latex implementation. If you get an error about missing .sty file, do the following:

  1. If you are an administrator on your computer, try to install the missing latex packages with your favorite package manager. The TeXLive or MiKTeX implementations should contain all required packages.

  2. If you are not an administrator, first get the missing packages, e.g. from CTAN. Collect the missing files in one or more directories and set the environment variable TEXINPUTS to these directories in the same way as the PATH variable is typically set.

Making a PyPI release

To create a release on PyPI, first increment the version number of pySecDec in pyproject.toml, and update Then make sure you have a recent Singularity installed, and run


This will create a source distribuion file (dist/*.tar.gz) and several prebuild distribution files (dist/*.whl); it will also print the instructions on how to double-check the files with auditwheel and upload them to PyPI using Twine.

Note that an account with PyPI is needed for this. Please see the packaging python projects guide for details.

The reason for using Singularity here is because prebuilt distributions must be built inside one of the manylinux Docker images: this guarantees that the prebuilt libraries and programs will work on a wide range of Linux systems. If it wasn't for this requirement, then making a distibution would be as simple as running

$ make dist

and then uploading the distfiles in dist/:

$ twine upload dist/pySecDec-<version>.tar.gz
$ twine upload dist/pySecDec-<version>-<tag>.whl

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